Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as:
$$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n], \ \textrm{s.t.} \ \sqrt{I} = \sqrt{(f_1, \ldots, f_m)}\right\} $$
The arithmetical rank measures the least number of equations necessary to define the scheme associated to $I$ set-theoretically.
In the case $n=p^2$ and $I_{p-1}$ is the ideal of $(p-1)$-minors of the generic matrix: $$ \begin{pmatrix} X_1 & X_2 & \ldots & X_{p} \\ X_{p+1} & X_{p+2} & \ldots & X_{2p} \\ \vdots & \vdots &\ldots & \vdots \\ X_{p^2-p+1} & X_{p^2-p+2} & \ldots & X_{p^2} \end{pmatrix}$$ the it is known that $ark(I_{p-1}) = 2p$.
The inequality $ark(I_{p-1})\leq 2p$ is not too difficult to prove and is a consequence, for instance, of the Desnanot-Jacobi identity. The only proof I know of the inequality $ark(I_{p-1})\geq 2p$ is due to Bruns and Schwänzl (generalizing an idea of Newstead) and uses étale (or complex analytic) cohomology.
I am interested in a sligtly cruder invariant, namely, let me denote:
$$ ark_r(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]^{(\leq r)}, \ \textrm{s.t.} \ \sqrt{I} = \sqrt{(f_1, \ldots, f_m)}\right\} $$ where $\mathbb{C}[X_1, \ldots X_n]^{(\leq r)}$ is the vector space of polynomials of total degree less or equal to $r$. My question is the following:
Question : Is there an elementary proof (that is, not using etale or complex analytic cohomology; let's say, a proof I could teach to a good master student having a first course in Algebraic Geometry) that: $$ ark_{p-1}(I_{p-1}) \geq 2p?$$
This inequality is an obvious consequence of the result of Bruns and Schwänzl I mentionned above. I am however hoping that adding the extra-condition that the $f_i$ should be of degree less or equal to the degree of a $p-1$-minor could make the inequality easier to prove via elementary representation theory/matrix theory/commutative algebra.
